The Halachot of Shabbat prohibit performing an act known as a "P’sik Resheh"- a permitted action that causes an inevitable automatic violation of Shabbat. The classic example is dragging a heavy chair on the ground. While dragging chairs is in itself a permitted act, nevertheless the chair will inevitably create a furrow in the ground, constituting a Melacha of Horesh. Even though the person’s intent was to bring the chair to the porch, and he had no interest whatsoever in creating a furrow, the act is prohibited.
A contemporary application of P’sik Resheh is opening a refrigerator with a light inside. Even though opening the door in and of itself is a permitted act, and his intent is to get food and not to turn on the light, it is prohibited, since the inevitable outcome of opening the door causes a Shabbat violation. As the Gemara explains, in such cases the outcome of turning on the light is treated as though that was his intent.
The question is whether one can instruct a non-Jew to open a refrigerator on Shabbat. Does the Halacha of "P’sik Resheh" also apply to a non-Jew? The Halacha states that there is no problem of P’sik Resheh with a non-Jew. It is permitted to tell him to open the refrigerator, despite the outcome. It is also permissible to instruct him to close the refrigerator. However, ideally, one should tell him to remove the light bulb while the door is still open. Removing the light bulb is only a Rabbinic prohibition, and therefore the act would constitute a "Shevut D’Shvut" (a double Rabbinic prohibition) which is permitted in extenuating circumstances like this, where one needs the food for his Shabbat meal. This way, he won’t have to call the non-Jew each time he needs something from the refrigerator.
It is permissible to instruct a non-Jew to open a refrigerator door on Shabbat even if a light will go on.